Starting off, I am in no way an expert and I will merely give my opinion as I have completed similar activities.
Starting off, you have to realize that there is either burning fat or gaining muscle, and it's very hard to do both at the same time without the use of something synthetic. The main reasoning for this is because if you want to gain muscular size and strength you really have to put yourself in a calorific surplus and if you want to lose fat you have to put yourself in a calorific deficit.
Additionally the goal of starting strength is going to be to gain strength and muscle. Whereas the goal from couch to 5k is going to be to burn fat and gain fitness, making them counter productive.
Also, you're going to have to listen to your body. If one day your taking your legs to fail doing squats and then the next day you're using them for fitness/running training, this can also be counter productive since your legs wont have recovered in time and they will now be over trained, especially if you're aiming to be in a caloric deficit.
Now that you have a bit more insight into the problems involved, there's a few different paths you can take. Firstly, you can continue doing the starting strength program and couch to 5k program if you feel that you are making solid gains in both areas, remember this is always about what your body is telling you.
Alternatively, I think you should ask yourself, what do you want the most? Depending on what you want more, you should focus your training more towards that style of training and limit yourself to the other style of training.
If you do still want both styles equally, a good style of program which balances the strength/fitness combination is something like a 100m-200m sprinters program. You can obviously edit the program to your liking, for example they will train a lot of things which assist in their running time such as high knees which you wouldn't need to train.
Also they will only typically aim at sprinting 100m for training, which you may want to extend - but the sprinters frame is quite muscular still, because they do a lot of things like deadlifts to help their explosive power. It is this combination of fitness/strength that you will be aiming for so that your training isn't being counter productive - so that it allows muscles to recover properly and is shaped around one goal instead two opposite goals.
Again, I am in no way an expert in the matter and I in no way know your body better you do - remember everything said is also a guideline, not a solid fact and the guidelines may be broken. Lastly, you'll probably find this out the hard way but running in general is a mental battle - never give up before you have to otherwise you're only cheating yourself. The best way to train past that mental barrier is by finding a training partner (a positive training partner!) that will push you past where your mind allows. Using this method will get you to your goal a lot faster than what you typically would find.